1. Copyright © 2006 The McGraw-Hill Companies, Inc. Programming Languages 2nd edition Tucker and Noonan Chapter 15 Logic Programming Q: How many legs does a dog have if you call its tail a leg? A: Four. Calling a tail a leg doesn’t make it one. Abraham Lincoln

2. Copyright © 2006 The McGraw-Hill Companies, Inc. Contents 15.1 Logic and Horn Clauses 15.2 Logic Programming in Prolog 15.2.1 Prolog Program Elements 15.2.2 Practical Aspects of Prolog 15.3 Prolog Examples 15.3.1 Symbolic Differentiation 15.3.2 Solving Word Puzzles 15.3.3 Natural Language Processing 15.3.4 Semantics of Clite 15.3 5 Eight Queens Problem

3. Copyright © 2006 The McGraw-Hill Companies, Inc. 15.1 Logic and Horn Clauses A Horn clause has a head h, which is a predicate, and a body, which is a list of predicates p 1, p 2, …, p n. It is written as: h  p 1, p 2, …, p n This means, “h is true only if p 1, p 2, …, and p n are simultaneously true.” E.g., the Horn clause: snowing(C)  precipitation(C), freezing(C) says, “it is snowing in city C only if there is precipitation in city C and it is freezing in city C.”

4. Copyright © 2006 The McGraw-Hill Companies, Inc. Horn Clauses and Predicates Any Horn clause h  p 1, p 2, …, p n can be written as a predicate: p 1  p 2  …  p n  h or equivalently:  (p 1  p 2  …  p n )  h But not every predicate can be written as a Horn clause. E.g.,literate(x)  reads(x)  writes(x)

5. Copyright © 2006 The McGraw-Hill Companies, Inc. Resolution and Unification If h is the head of a Horn clause h  terms and it matches one of the terms of another Horn clause: t  t 1, h, t 2 then that term can be replaced by h’s terms to form: t  t 1, terms, t 2 During resolution, assignment of variables to values is called instantiation. Unification is a pattern-matching process that determines what particular instantiations can be made to variables during a series of resolutions.

6. Copyright © 2006 The McGraw-Hill Companies, Inc. Example The two clauses: speaks(Mary, English) talkswith(X, Y)  speaks(X, L), speaks(Y, L), X  Y can resolve to: talkswith(Mary, Y)  speaks(Mary, English), speaks(Y, English), Mary  Y The assignment of values Mary and English to the variables X and L is an instantiation for which this resolution can be made.

7. Copyright © 2006 The McGraw-Hill Companies, Inc. 15.2 Logic Programming in Prolog In logic programming the program declares the goals of the computation, not the method for achieving them. Logic programming has applications in AI and databases. –Natural language processing (NLP) –Automated reasoning and theorem proving –Expert systems (e.g., MYCIN) –Database searching, as in SQL (Structured Query Language) Prolog emerged in the 1970s. Distinguishing features: –Nondeterminism –Backtracking

8. Copyright © 2006 The McGraw-Hill Companies, Inc. 15.2.1 Prolog Program Elements Prolog programs are made from terms, which can be: –Variables –Constants –Structures Variables begin with a capital letter, like Bob. Constants are either integers, like 24, or atoms, like the, zebra, ‘Bob’, and ‘.’. Structures are predicates with arguments, like: n(zebra ), speaks(Y, English ), and np(X, Y) –The arity of a structure is its number of arguments (1, 2, and 2 for these examples).

9. Copyright © 2006 The McGraw-Hill Companies, Inc. Facts, Rules, and Programs A Prolog fact is a Horn clause without a right-hand side. Its form is (note the required period. ): term. A Prolog rule is a Horn clause with a right-hand side. Its form is (note :- represents  and the required period. ): term :- term 1, term 2, … term n. A Prolog program is a collection of facts and rules.

10. Copyright © 2006 The McGraw-Hill Companies, Inc. Example Program speaks(allen, russian). speaks(bob, english). speaks(mary, russian). speaks(mary, english). talkswith(X, Y) :- speaks(X, L), speaks(Y, L), X \= Y. This program has four facts and one rule. The rule succeeds for any instantiation of its variables in which all the terms on the right of := are simultaneously true. E.g., this rule succeeds for the instantiation X=allen, Y=mary, and L=russian. For other instantiations, like X=allen and Y=bob, the rule fails.

11. Copyright © 2006 The McGraw-Hill Companies, Inc. Searching for Success: Queries A query is a fact or rule that initiates a search for success in a Prolog program. It specifies a search goal by naming variables that are of interest. E.g., ?- speaks(Who, russian). asks for an instantiation of the variable Who for which the query speaks(Who, russian) succeeds. A program is loaded by the query consult, whose argument names the program. E.g., ?- consult(speaks). loads the program named speaks, given on the previous slide.

12. Copyright © 2006 The McGraw-Hill Companies, Inc. Answering the Query: Unification To answer the query: ?- speaks(Who, russian). Prolog considers every fact and rule whose head is speaks. (If more than one, consider them in order.) Resolution and unification locate all the successes: Who = allen ; Who = mary ; No –Each semicolon ( ; ) asks, “Show me the next success.”

13. Copyright © 2006 The McGraw-Hill Companies, Inc. Search Trees First attempt to satisfy the query ?- talkswith(Who, allen). Fig 15.2

14. Copyright © 2006 The McGraw-Hill Companies, Inc. Database Search - The Family Tree Fig 15.4

15. Copyright © 2006 The McGraw-Hill Companies, Inc. Prolog Program Fig 15.3 mother(mary, sue). mother(mary, bill). mother(sue, nancy). mother(sue, jeff). mother(jane, ron). parent(A,B) :- father(A,B). parent(A,B) :- mother(A,B). grandparent(C,D) :- parent(C,E), parent(E,D). father(john, sue). father(john, bill). father(bob, nancy). father(bob, jeff). father(bill, ron).

16. Copyright © 2006 The McGraw-Hill Companies, Inc. Some Database Queries Who are the parents of jeff? ?- parent(Who, jeff). Who = bob; Who = sue Find all the grandparents of Ron. ?- grandparent(Who, ron). What about siblings? Those are the pairs who have the same parents. ?- sibling(X, Y) :- parent(W, X), parent(W, Y), X\=Y.

17. Copyright © 2006 The McGraw-Hill Companies, Inc. Lists A list is a series of terms separated by commas and enclosed in brackets. –The empty list is written []. –The sentence “The giraffe dreams” can be written as a list: [the, giraffe, dreams] –A “don’t care” entry is signified by _, as in [_, X, Y] –A list can also be written in the form: [Head | Tail] –The functions append joins two lists, and member tests for list membership.

18. Copyright © 2006 The McGraw-Hill Companies, Inc. append Function append([], X, X). append([Head | Tail], Y, [Head | Z]) :- append(Tail, Y, Z). This definition says: 1. Appending the empty list to any list ( X ) returns an unchanged list ( X again). 2. If Tail is appended to Y to get Z, then a list one element larger [Head | Tail] can be appended to Y to get [Head | Z]. Note: The last parameter designates the result of the function. So a variable must be passed as an argument.

19. Copyright © 2006 The McGraw-Hill Companies, Inc. member Function member(X, [X | _]). member(X, [_ | Y]) :- member(X, Y). The test for membership succeeds if either: 1. X is the head of the list [X | _] 2. X is not the head of the list [_ | Y], but X is a member of the list Y. Notes: pattern matching governs tests for equality. Don’t care entries ( _ ) mark parts of a list that aren’t important to the rule.

20. Copyright © 2006 The McGraw-Hill Companies, Inc. More List Functions X is a prefix of Z if there is a list Y that can be appended to X to make Z. That is: prefix(X, Z) :- append(X, Y, Z). Similarly, Y is a suffix of Z if there is a list X to which Y can be appended to make Z. That is: suffix(Y, Z) :- append(X, Y, Z). So finding all the prefixes (suffixes) of a list is easy. E.g.: ?- prefix(X, [my, dog, has, fleas]). X = []; X = [my]; X = [my, dog]; …